In performing performance attribution, the returns of a portfolio are compared against those of a benchmark, and the excess return (i.e., relative performance) is attributed to various effects resulting from active decisions by the portfolio managers. Performance attribution is a rich and complex topic, which can be viewed from many angles. There are a variety of conventional methods for performing attribution based on a single-period analysis. However, if performance is measured over an extended length of time, a single-period buy-and-hold analysis may lead to significant errors, especially for highly active portfolios. Therefore, it is imperative to link the single-period attribution effects over multiple periods in an accurate and meaningful way. The two basic approaches that have arisen for such linking are the arithmetic and geometric methodologies.
In arithmetic attribution, the performance of a portfolio relative to a benchmark is given by the difference R− R, where R and R refer to portfolio and benchmark returns, respectively. This relative performance, in turn, is decomposed sector by sector into attribution effects that measure how well the portfolio manager weighted the appropriate sectors and selected securities within the sectors. The sum of the attribution effects gives the performance, R− R.
In geometric attribution, by contrast, the relative performance is defined by the ratio (1+R)/(1+ R). This relative performance is again decomposed sector by sector into attribution effects. In this case, however, it is the product of the attribution effects that gives the relative performance (1+R)/(1+ R). A recent example of both arithmetic and geometric attribution systems is described in Carino, “Combining Attribution Effects Over Time,” Journal of performance Measurement, Summer 1999, pp. 5–14 (“Carino”).
An advantage of the arithmetic approach is that it is more intuitive. For instance, if the portfolio return is 21% and the benchmark return is 10%, most people regard the relative performance to be 11%, as opposed to 10%. An advantage of geometric attribution, on the other hand, is the ease with which attribution effects can be linked over time.
Carino describes one possible algorithm for linking attribution effects over time that results in a multi-period arithmetic performance attribution system. Furthermore, the result is residual free in that the sum of the linked attribution effects is exactly equal to the difference in linked returns. Carino discloses an arithmetic performance attribution method which determines portfolio relative performance over multiple time periods as a sum of terms of form (Rt− Rt)βt, where the index “t” indicates one time period, and where Carino's coefficients βt are
      β    t    Carino    =            [                        R          -                      R            _                                                ln            ⁢                          (                              1                +                R                            )                                -                      ln            ⁢                          (                              1                +                                  R                  _                                            )                                          ]        ⁢                  (                                            ln              ⁢                              (                                  1                  +                                      R                    t                                                  )                                      -                          ln              ⁢                              (                                  1                  +                                                            R                      _                                        t                                                  )                                                                        R              t                        -                                          R                _                            t                                      )            .      
In accordance with the present invention, new coefficients to be defined below replace Carino's coefficients βt (sometimes referred to herein as conventional “logarithmic” coefficients). The inventive coefficients have a much smaller standard deviation than the conventional logarithmic coefficients and are metric preserving. Reducing the standard deviation of the coefficients is important in order to minimize the distortion that arises from overweighting certain periods relative to others.